Cantor Confusion
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From: cbrown on 6 Dec 2006 20:34 Dik T. Winter wrote: > In article <1165369900.708031.124650(a)n67g2000cwd.googlegroups.com> cbrown(a)cbrownsystems.com writes: > > Dik T. Winter wrote: > ... > > > Sorry, I can't read Word documents (I have no reader available). > > > But again who requires one-point compactification with what goal? > > > > His paper "Adaptation of Spectral Analysis to Reality" doesn't contain > > the word "compactification" (or even the fragment "compact"). It seems > > to promote the use of only the real parts of a Fourier Transform > > (because complex values have no "physical reality"; even negative > > values are suspect at best) and restricting the time domain to be (-oo, > > 0] (because we only know the past and cannot claim to know the future). > > You mean he considers negative numbers as suspect but does allow only > negative numbers for the time domain? Ah, but he rectifies this by discarding the isin(wt) aspect of the transform, and retaining only the cos(wt) part. Since cos(wt) is symmetric, he then substitutes -t for t; and integrates over [0, oo+); thus his comments regarding R+. > And apparently he has never heard > about shifting the origin. He finds shifting the origin to be an arbitrary, unrealistic activity; which may introduce errors. Thus his "elapsed time" versus "usual time". > According to that reasoning we are forever > living in the year 0, while the coordinates of all past years go up by > one on the first of January every year. And by the same reasoning we > can not even state that on the first of January 2007 Slovenia will > introduce the euro, because we can not look in the future. Bizarre. > It has some pretty pictures, though. Cheers - Chas
From: Dik T. Winter on 6 Dec 2006 21:26 In article <1165453290.153284.236070(a)73g2000cwn.googlegroups.com> "David R Tribble" <david(a)tribble.com> writes: .... > > SUBROUTINE CHANGE(A, B) > > IF(A .EQ. 25) A = B > > RETURN > > END > > > > Now try: > > CALL CHANGE(25, 17) > > But I still don't see how calling a subroutine changes the value of 25. > Unless you have a memory overwrite bug. Oh, it did work on systems that had no write protected memory and that just passed the address of the constant to the subroutine. > A better programming metaphor is that numbers and sets are constants, > not variables. Remember the programmers paradigm: "constants aren't and variables won't". -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 6 Dec 2006 22:10 In article <4576D4C7.4070207(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 10:41 PM, Virgil wrote: > > In article <457580FA.1030700(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> That there are more rationals than reals? > > > > WE do not make any such assumption. > > I perhaps meant more reals than rationals. I apologize. > > Dedekind as well as Cantor started from this idea. Cantor even > fabricated the notion cardinality in order to quantify the putative > difference in size. Considering the rationals a subset of the reals, > dull people conclude that there must be more reals than rationals. There certainly cannot be fewer, or even just as many, so what other option is open/
From: Virgil on 6 Dec 2006 22:18 In article <4576D8A1.1030807(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 10:54 PM, Virgil wrote: > > In article <4575A562.60602(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > > > >> Blissful ignorance of mathematicians does not utter complains if the > >> axiom of (possibly infinite) extensionality claims the existence of a > >> set which has to include all of its elements. > > > > EB should read what axioms of extensionality actually says before > > pontificating. > > Roughly speaking, it just claims that a set is unambiguously determined > by its elements. If i recall correctly A=B<-->(A in B and B in A) Then it does not state the truism that a set must contain all its elements as EB originally implied.
From: Virgil on 6 Dec 2006 22:32
In article <4576DCFC.6040901(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 10:59 PM, Virgil wrote: > > In article <4575B447.50208(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/5/2006 2:13 PM, Bob Kolker wrote: > >> > Eckard Blumschein wrote: > >> >> an exact numerical representation available. Kronecker said, they are no > >> >> numbers at all. Since the properties of the reals have to be the same as > >> >> these of the irrationals, all reals must necessarily also be uncountable > >> >> fictions. > >> > > >> > For the latest time. Uncountability is a property of sets, not > >> > individual numbers. > >> > >> I know this widespread view. > >> > >> > There is no such thing as an uncountable real > >> > number. > >> > >> Real numbers according to DA2 are uncountable altogether. People like > >> you will not grasp that. Not a single real number is countable. > >> > >> > Nor is there any such thing as a countable integer. > >> > >> Every single integer is a countable element. > > > > Since the set of integers is a subset of the set of reals, every integer > > is a real in any real set theory, whatever EB may claim. > > Just this is fallacious. The reals are continuous like glue. Being like glue is not a mathematically relevant quality. The Dedekind cut for a real, e.g., for sqrt(2), is as actual as is the rational for 1/2. > Embedded genuine numbers lost their numerical address Stupid of them. > > > >> > >> > >> > Countability > >> > /Uncountability are properties of -sets-, not individuals. > >> > >> Do not reiterate what I know but deny. > > > > Then do not reiterate your falsehoods > > Cantor's proof (DA2) of uncountability required numbers of perfectly > infinite length. Cantor's first proof did not require anything but one of the extant models of the reals within set theory. No decimal or other basal representation was required. Note that the Dedekind cut is only a mere refinement of Eudoxus treatment of incommeasurables which has been acceptable since the time of Euclid. You understand: Such numbers contradict common sense. Common sense is irrelevant in mathematics. > Writing very fasr and without proofreading, I cannot guarantee correct > command of my English. Sometimes words or letters may be missing or > confused. Nevertheless, all these arguments of mine are most likely > flawless. Your flaws in English are excusable, your flaws in logic are not. > > > >> > You have been told this on several occassions and you apparently are too > >> > stupid to learn it. > >> > >> You should try and refute. > > > > We have refuted it many times, but those who choose not to see remain > > blind. > > You did not refute a single argument of mine. That you have not recognized that you have been refuted is one of your faults. |